# 722edo

← 721edo | 722edo | 723edo → |

^{2}**722 equal divisions of the octave** (abbreviated **722edo** or **722ed2**), also called **722-tone equal temperament** (**722tet**) or **722 equal temperament** (**722et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 722 equal parts of about 1.66 ¢ each. Each step represents a frequency ratio of 2^{1/722}, or the 722nd root of 2.

722edo is a strong 2.7.19.23 subgroup tuning, with 179\722 being a semiconvergent to the log_{2}(19/16). Despite having a strong approximation of 7, it is only consistent upwards to the 5-limit.

Using the 421\722 fifth, it supports a variant of fifth-stacked tuning that divides 38th harmonic into 9 parts, meaning that C - D# in this system is equal to 19/16, the otonal minor third. This creates a peculiar violation of Western theory which would require spelling this minor triad involving 19/16 as C-D#-G instead of C-Eb-G. This can be realized as 355 & 722 2.17.19.23 temperament from a regular temperament theory perspective - it should be noted that the fifth is not mapped to 3/2 but is slightly flatter.

Aside from this, 722bc val tempers out the hemifamity comma and is a tuning for the undecental temperament. Since 722 is divisible by 19, the 722dg val is a tuning for the kalium temperament in the 19-limit.

### Odd harmonics

Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | -0.570 | -0.718 | +0.149 | +0.522 | +0.483 | +0.470 | +0.374 | -0.246 | -0.006 | -0.421 | -0.019 |

Relative (%) | -34.3 | -43.2 | +9.0 | +31.4 | +29.0 | +28.3 | +22.5 | -14.8 | -0.4 | -25.3 | -1.2 | |

Steps (reduced) |
1144 (422) |
1676 (232) |
2027 (583) |
2289 (123) |
2498 (332) |
2672 (506) |
2821 (655) |
2951 (63) |
3067 (179) |
3171 (283) |
3266 (378) |